what is the broken data source? something like a basemap? a file on a server? Have you tried creating a new dataframe, moving all the good files into it, then deleting/removing the data frame with the broken link in it? that way you don't have to touch the file, if that is what is causing the problem.
I have an addin /blogs/myAlaskaGIS/2015/08/31/python-addin-for-data-inventory-and-broken-link-repair?sr=search&searc... that may help you find and repair the broken link. If nothing else, the tool to list the broken links is handy.
Also, in ArcCatalog, you should be able to right-click on the mxd and choose fix datasets. This creates a new mxd, so it doesn't change the mxd. You have to know what it is you are wanting to change for the tool, but maybe even saving it without changing will help.